If $f : [a, b] \to (0,\infty)$; $0 < a \leq b$; $f$ derivable; $f'$ continuous then:$$\int \limits_{a}^{b}\frac{f'(x)\sqrt{f(x)}}{f^3(x) + 1}\leq \tan^{-1}\left(\frac{f(b)-f(a)}{1 + f(a)f(b)}\right)$$

Let $x$ be in the interval $\left(0, \frac{\pi}{2}\right)$ such that $\sin x - \cos x = \frac12$ . Then $\sin^3 x + \cos^3 x = \frac{m\sqrt{p}}{n}$ , where $m, n$, and $p$ are relatively prime positive integers, and $p$ is not divisible by the square of any prime. Find $m + n + p$.

Let $ABC$ be a right triangle where $\measuredangle A = 90^\circ$ and $M\in (AB)$ such that $\frac{AM}{MB}=3\sqrt{3}-4$. It is known that the symmetric point of $M$with respect to the line $GI$ lies on $AC$. Find the measure of $\measuredangle B$.

In rhombus $NIMO$, $MN = 150\sqrt{3}$ and $\measuredangle MON = 60^{\circ}$. Denote by $S$ the locus of points $P$ in the interior of $NIMO$ such that $\angle MPO \cong \angle NPO$. Find the greatest integer not exceeding the perimeter of $S$. [i]Proposed by Evan Chen[/i]

Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75 $

In $\triangle ABC$, the corresponding sides of angle $A,B,C$ are $a,b,c$ respectively. If $a\cos B-b\cos A=\frac{3}{5}c$, find the value of $\frac{\tan A}{\tan B}$.

Consider the points of intersection of the graphs $y = \cos x$ and $x = 100 \cos (100y)$ for which both coordinates are positive. Let $a$ be the sum of their $x$-coordinates and $b$ be the sum of their $y$-coordinates. Determine the value of $\frac{a}{b}$.

Evaluate $ \int_0^{\frac {\pi}{6}} \frac {\sin x \plus{} \cos x}{1 \minus{} \sin 2x}\ln\ (2 \plus{} \sin 2x)\ dx.$

Let $f$ be the function defined by $f(x) = 4x(1 - x)$. Let $n$ be a positive integer. Prove that there exist distinct real numbers $x_1$, $x_2$, $\ldots\,$, $x_n$ such that $x_{i + 1} = f(x_i)$ for each integer $i$ with $1 \le i \le n - 1$, and such that $x_1 = f(x_n)$.

Find all positive integer $n$ such that $$\frac{\sin{n\theta}}{\sin{\theta}} - \frac{\cos{n\theta}}{\cos{\theta}} = n-1$$ holds for all $\theta$ which are not integral multiples of $\frac{\pi}{2}$

Let $\triangle ABC$ be an acute triangle and $AD$ the bisector of the angle $\angle BAC$ with $D\in(BC)$. Let $E$ and $F$ denote feet of perpendiculars from $D$ to $AB$ and $AC$ respectively. If $BF\cap CE=K$ and $\odot AKE\cap BF=L$ prove that $DL\perp BF$.

An infinitely long slab of glass is rotated. A light ray is pointed at the slab such that the ray is kept horizontal. If $\theta$ is the angle the slab makes with the vertical axis, then $\theta$ is changing as per the function \[ \theta(t) = t^2, \]where $\theta$ is in radians. Let the $\emph{glassious ray}$ be the ray that represents the path of the refracted light in the glass, as shown in the figure. Let $\alpha$ be the angle the glassious ray makes with the horizontal. When $\theta=30^\circ$, what is the rate of change of $\alpha$, with respect to time? Express your answer in radians per second (rad/s) to 3 significant figures. Assume the index of refraction of glass to be $1.50$. Note: the second figure shows the incoming ray and the glassious ray in cyan. [asy] fill((6,-2-sqrt(3))--(3, -2)--(3+4/sqrt(3),2)--(6+4/sqrt(3),2-sqrt(3))--cycle, gray(0.7)); draw((3+2/sqrt(3),0)--(10,0),linetype("4 4")+grey); draw((0,0)--(3+2/sqrt(3),0)); draw((3+2/sqrt(3), 0)--(7+2/sqrt(3), -1)); arrow((6.5+2/sqrt(3), -7/8), dir(180-14.04), 9); draw((3,-2)--(3,2), linetype("4 4")); draw((6,-2-sqrt(3))--(3,-2)--(3+2/sqrt(3),0)--(3+4/sqrt(3), 2)--(6+4/sqrt(3), 2-sqrt(3))); draw(anglemark((3+2/sqrt(3),0),(3,-2),(3,0),15)); label("$\theta$", (3.2, -1.6), N, fontsize(8)); label("$\alpha$", (6, -0.2), fontsize(8)); [/asy] [asy] fill((6,-2-sqrt(3))--(3, -2)--(3+4/sqrt(3),2)--(6+4/sqrt(3),2-sqrt(3))--cycle, gray(0.7)); draw((3+2/sqrt(3),0)--(10,0),linetype("4 4")+grey); draw((0,0)--(3+2/sqrt(3),0), cyan); draw((3+2/sqrt(3), 0)--(7+2/sqrt(3), -1), cyan); arrow((6.5+2/sqrt(3), -7/8), dir(180-14.04), 9, cyan); draw((3,-2)--(3,2), linetype("4 4")); draw((6,-2-sqrt(3))--(3,-2)--(3+2/sqrt(3),0)--(3+4/sqrt(3), 2)--(6+4/sqrt(3), 2-sqrt(3))); draw(anglemark((3+2/sqrt(3),0),(3,-2),(3,0),15)); label("$\theta$", (3.2, -1.6), N, fontsize(8)); label("$\alpha$", (6, -0.2), fontsize(8)); [/asy] [i]Problem proposed by Ahaan Rungta[/i]

If the number of triangles that $\angle ABC=60^{\circ},AC=12,BC=k$ is exactly one, then the range value of $k$ is $\text{(A)}k=8\sqrt3\qquad\text{(B)}k=0<k\leq12\qquad\text{(C)}k\geq12\qquad\text{(D)}k=8\sqrt3\text { or }0<k\leq12$

Given an acute triangle $ PBC$ with $ PB\neq PC.$ Points $ A,D$ lie on $ PB,PC,$ respectively. $ AC$ intersects $ BD$ at point $ O.$ Let $ E,F$ be the feet of perpendiculars from $ O$ to $ AB,CD,$ respectively. Denote by $ M,N$ the midpoints of $ BC,AD.$ $ (1)$: If four points $ A,B,C,D$ lie on one circle, then $ EM\cdot FN \equal{} EN\cdot FM.$ $ (2)$: Determine whether the converse of $ (1)$ is true or not, justify your answer.

The graph of $r=2+\cos2\theta$ and its reflection over the line $y=x$ bound five regions in the plane. Find the area of the region containing the origin.

At what smallest $n$ is there a convex $n$-gon for which the sines of all angles are equal and the lengths of all sides are different?

Find the continuous function $f(x)$ such that the following equation holds for any real number $x$. \[\int_0^x \sin t \cdot f(x-t)dt=f(x)-\sin x.\] [i]1977 Keio University entrance exam/Medicine[/i]

Suppose $ABCD$ is a square piece of cardboard with side length $a$. On a plane are two parallel lines $\ell_1$ and $\ell_2$, which are also $a$ units apart. The square $ABCD$ is placed on the plane so that sides $AB$ and $AD$ intersect $\ell_1$ at $E$ and $F$ respectively. Also, sides $CB$ and $CD$ intersect $\ell_2$ at $G$ and $H$ respectively. Let the perimeters of $\triangle AEF$ and $\triangle CGH$ be $m_1$ and $m_2$ respectively. Prove that no matter how the square was placed, $m_1+m_2$ remains constant.

Consider a circle $\Gamma$, a point $A$ on its exterior, and the points of tangency $B$ and $C$ from $A$ to $\Gamma$. Let $P$ be a point on the segment $AB$, distinct from $A$ and $B$, and let $Q$ be the point on $AC$ such that $PQ$ is tangent to $\Gamma$. Points $R$ and $S$ are on lines $AB$ and $AC$, respectively, such that $PQ\parallel RS$ and $RS$ is tangent to $\Gamma$ as well. Prove that $[APQ]\cdot[ARS]$ does not depend on the placement of point $P$.

The point $P$ lies inside, or on the boundary of, the triangle $ABC$. Denote by $d_{a}$, $d_{b}$ and $d_{c}$ the distances between $P$ and $BC$, $CA$, and $AB$, respectively. Prove that $\max\{AP,BP,CP \} \ge \sqrt{d_{a}^{2}+d_{b}^{2}+d_{c}^{2}}$. When does the equality holds?

Let $ABC$ be an acute-angled triangle. The external bisector of $\angle{BAC}$ meets the line $BC$ at point $N$. Let $M$ be the midpoint of $BC$. $P$ and $Q$ are two points on line $AN$ such that, $\angle{PMN}=\angle{MQN}=90^{\circ}$. If $PN=5$ and $BC=3$, then the length of $QA$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are coprime positive integers. What is the value of $(a+b)$?

Let $ABC$ be a triangle with $\angle A=60^{\circ}$ and $AB\neq AC$, $I$-incenter, $O$-circumcenter. Prove that perpendicular bisector of $AI$, line $OI$ and line $BC$ have a common point.

Let $AB$ and $BC$ be two consecutive sides of a regular polygon with 9 vertices inscribed in a circle of center $O$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of the radius perpendicular to $BC$. Find the measure of the angle $\angle OMN$.

Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to $ 32$? $ \textbf{(A)}\ 560 \qquad \textbf{(B)}\ 564 \qquad \textbf{(C)}\ 568 \qquad \textbf{(D)}\ 1498 \qquad \textbf{(E)}\ 2255$

Let $ABC$ be an acute triangle and $H$ is orthocenter. Let $D$ be the intersection of $BH$ and $AC$ and $E$ be the intersection of $CH$ and $AB$. The circumcircle of $ADE$ cuts the circumcircle of $ABC$ at $F \neq A$. Prove that the angle bisectors of $\angle BFC$ and $\angle BHC$ concur at a point on $BC.$